Find Intersection of Two Non-planar Surfaces

Most geologic structures are not ideally planar.
Nevertheless, we may still have to locate the intersections of,
say, a folded bed or an intrusive contact with a curving fault.
One particularly important case of two intersecting surfaces is
the intersection of any geologic structure with the topography. For now, let us consider the intersection of
relatively simple surfaces.

Most of the methods we used in finding the intersections of
two planes still apply; after all, planes are surfaces. The
structure contours are not straight lines and the intersection is
not a straight line, but in the most important respects, the two
situations are the same. We find the intersections of matching
structure contours and draw a smooth curve through the
intersection points.

Since the spacing of contours varies, it may be useful to
interpolate intermediate points.

A Few Cardinal Rules

An intersection line can never cross one contour unless the
same elevation contour from the other surface intersects at that
point. You must have three lines crossing: the intersection line and two contour lines.

Be extremely careful that you locate the intersections of
matching contours. It can be very easy to mismatch contours.

Example: Simple Surfaces

1. Find the intersection of these two dikes with their
structure contours shown.

2. Locate the intersections of matching contours.

3. Interpolate where contours vary greatly in spacing.

4. Construct a smooth curve through the intersections.

More Complex Problems

Problems such as the intersection of three surfaces or the
intersection of a line and a surface can be solved in much the
same way as the simple straight line and plane problems by using
the intersection method described here.

Geologic structures in the subsurface can result in situations that are
unfamiliar. Structure contours can appear to cross (structures may have overhangs or
layers can be overturned) and they can terminate (at faults). Structure
contours only appear to cross because a 500-meter contour that crosses a
400-meter contour is actually 100 meters above it.

Example: Complex Surfaces

1. A fold with structure contours. The fold is overturned.
It is actually recumbent and cannot be considered either an anticline or a
syncline.

2. A small intrusion that intersects the fold. It is shown here by
itself. The intrusion is shaped somewhat like an upside-down pear tilted
to the north.

3. Here the top surface of the fold is colored to help visualize its
form.

4. The top surface of the intrusion is similarly colored.

Note that the contours appear to cross. That is, their projections on the
surface cross. The contours themselves are separated vertically by 100 meter
intervals.

In a case like this you do whatever you have to do to visualize the structure.
It may mean making cross-sections or other auxiliary techniques. You may end up
building three-dimensional models if the stakes are high enough. Whatever it
takes.

This is a fairly nasty case. A couple of comments are in order. First of all,
where the intrusion cuts the fold, obviously that part of the fold no longer exists.
As we do the construction we treat the fold and the intrusion as
interpenetrating mathematical surfaces, but physically that is not the case. We might assume
the structure contours on the fold were generated by extrapolating surface data,
perhaps in conjunction with borehole information. The shape of the intrusion
might be known from surface outcrop, borehole data, geophysical data, or a
combination of all of them. Perhaps the contact between the intrusion and the
contoured unit in the fold is mineralized, creating an economic reason to map
the intersection. Geologists analyzing such a case might spend weeks on the
problem.

Analyzing Structural Levels

Here we have shown the relationships at each contour. Yellow
shows the part of the intrusion within the fold, magenta shows the portion
outside. The other structure contours are shown subdued for reference.

At 200 meters the intrusion cuts the northeast (lower) limb of the
fold.

At 300 meters it just reaches the southwest (upper) limb.

At 400 and 500 meters it encloses the entire hinge of the fold.

The intrusion does not reach 600 meters at all.

Cross-Sections

Here we draw cross-sections. The cross-section lines are
shown in red on the map. The section below the map is the E-W section,
that to the right is the N-S section. Note the 2x vertical exaggeration.

The diagonal reflection line is a common technique in drafting for
transferring dimensions through 90 degrees.

The left cross section shows the intrusion enclosing the entire hinge
of the fold. On the right the intrusion penetrates through the fold.

In the left diagram the fold appears to be an anticline, but in the right the
limbs converge gradually downward and it looks like a syncline! (We can
see on the map that the cross-section line will eventually cross the hinge to
the south). Paradoxes like this are common with steeply-plunging recumbent
folds.

Based on these two analyses, we expect the intersection will be a
saddle-shaped curve wrapping around the hinge of the fold. Picture taking a bite
out of a taco shell. Now to construct it.

Constructing the Intersection

1. Structure contours on the fold (green) and intrusion
(blue).

2. Identify intersections of like contours. Here they are color coded.

3. The line of points from 200 to 500 meters at the rear of the
intrusion is pretty straightforward. So are a couple of other pairs of
points. The intersections are shown in red.

4. The pluton does not extend down to 100 meters or up to 600, so the
200- and 500- meter points must connect as shown. Although the curve
appears to intersect the 600-meter contour, it is actually beneath it.

5. The intrusion and fold contours just graze at the
remaining 300- and 400- meter points, suggesting that the contact just
reaches those points and turns

6. The complete contact, but what does it mean?

7. The contours on the intrusion are subdued here. The surface of the
fold has been colored to illustrate the area enclosed by the contact:
yellow on the top surface of the fold, dark green on the back, light green
where they overlap.

8. From 7. we see the construction has a flaw - the contact does not
wrap around the hinge of the fold (dark gray). Modifying it as shown
remedies the problem.

You can see that there is no single recipe for doing this. The only way to do
it is to know as much geology as possible, visualize the intersecting surfaces
as completely as possible, and proceed from the simpler to the more complex
parts. Also, there is no best way to portray the solution. A map view might be
very confusing. It might need to be supplemented by cross-sections or
perspective views. Computer drafting programs can help enormously, especially if
they generate views that can be rotated, but they only
supplement geological intuition, not substitute for it.